How do Artificial Neural Networks store data compared to Biological Neural Networks?
Second question first: Data is stored in an ANN in the form of weights in the adjacency matrix between neurons. During training, these weights are updated by a learning algorithm (such as backpropagation).

First question: according to award-winning neuroscientist Tim Bliss:


  “It’s been accepted really since the turn of the 20th century, since the time of the Spanish neuroscientist Ramón y Cajal, that really the only place where memories can be stored is at synapses, the junctions between nerve cells.


A protein called the NMDA receptor plays a key roll in the strengthening of synaptic connections (which is more broadly achieved by a form of Hebbian Learning).